:mod:`numbers` --- Numeric abstract base classes
================================================
.. module:: numbers
:synopsis: Numeric abstract base classes (Complex, Real, Integral, etc.).
The :mod:`numbers` module (:pep:`3141`) defines a hierarchy of numeric
:term:`abstract base classes ` which progressively define
more operations. None of the types defined in this module can be instantiated.
.. class:: Number
The root of the numeric hierarchy. If you just want to check if an argument
*x* is a number, without caring what kind, use ``isinstance(x, Number)``.
The numeric tower
-----------------
.. class:: Complex
Subclasses of this type describe complex numbers and include the operations
that work on the built-in :class:`complex` type. These are: conversions to
:class:`complex` and :class:`bool`, :attr:`.real`, :attr:`.imag`, ``+``,
``-``, ``*``, ``/``, :func:`abs`, :meth:`conjugate`, ``==``, and ``!=``. All
except ``-`` and ``!=`` are abstract.
.. attribute:: real
Abstract. Retrieves the real component of this number.
.. attribute:: imag
Abstract. Retrieves the imaginary component of this number.
.. method:: conjugate()
Abstract. Returns the complex conjugate. For example, ``(1+3j).conjugate()
== (1-3j)``.
.. class:: Real
To :class:`Complex`, :class:`Real` adds the operations that work on real
numbers.
In short, those are: a conversion to :class:`float`, :func:`math.trunc`,
:func:`round`, :func:`math.floor`, :func:`math.ceil`, :func:`divmod`, ``//``,
``%``, ````, and ``>=``.
Real also provides defaults for :func:`complex`, :attr:`~Complex.real`,
:attr:`~Complex.imag`, and :meth:`~Complex.conjugate`.
.. class:: Rational
Subtypes :class:`Real` and adds
:attr:`~Rational.numerator` and :attr:`~Rational.denominator` properties, which
should be in lowest terms. With these, it provides a default for
:func:`float`.
.. attribute:: numerator
Abstract.
.. attribute:: denominator
Abstract.
.. class:: Integral
Subtypes :class:`Rational` and adds a conversion to :class:`int`. Provides
defaults for :func:`float`, :attr:`~Rational.numerator`, and
:attr:`~Rational.denominator`. Adds abstract methods for ``**`` and
bit-string operations: ``<>``, ``&``, ``^``, ``|``, ``~``.
Notes for type implementors
---------------------------
Implementors should be careful to make equal numbers equal and hash
them to the same values. This may be subtle if there are two different
extensions of the real numbers. For example, :class:`fractions.Fraction`
implements :func:`hash` as follows::
def __hash__(self):
if self.denominator == 1:
# Get integers right.
return hash(self.numerator)
# Expensive check, but definitely correct.
if self == float(self):
return hash(float(self))
else:
# Use tuple's hash to avoid a high collision rate on
# simple fractions.
return hash((self.numerator, self.denominator))
Adding More Numeric ABCs
~~~~~~~~~~~~~~~~~~~~~~~~
There are, of course, more possible ABCs for numbers, and this would
be a poor hierarchy if it precluded the possibility of adding
those. You can add ``MyFoo`` between :class:`Complex` and
:class:`Real` with::
class MyFoo(Complex): ...
MyFoo.register(Real)
Implementing the arithmetic operations
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
We want to implement the arithmetic operations so that mixed-mode
operations either call an implementation whose author knew about the
types of both arguments, or convert both to the nearest built in type
and do the operation there. For subtypes of :class:`Integral`, this
means that :meth:`__add__` and :meth:`__radd__` should be defined as::
class MyIntegral(Integral):
def __add__(self, other):
if isinstance(other, MyIntegral):
return do_my_adding_stuff(self, other)
elif isinstance(other, OtherTypeIKnowAbout):
return do_my_other_adding_stuff(self, other)
else:
return NotImplemented
def __radd__(self, other):
if isinstance(other, MyIntegral):
return do_my_adding_stuff(other, self)
elif isinstance(other, OtherTypeIKnowAbout):
return do_my_other_adding_stuff(other, self)
elif isinstance(other, Integral):
return int(other) + int(self)
elif isinstance(other, Real):
return float(other) + float(self)
elif isinstance(other, Complex):
return complex(other) + complex(self)
else:
return NotImplemented
There are 5 different cases for a mixed-type operation on subclasses
of :class:`Complex`. I'll refer to all of the above code that doesn't
refer to ``MyIntegral`` and ``OtherTypeIKnowAbout`` as
"boilerplate". ``a`` will be an instance of ``A``, which is a subtype
of :class:`Complex` (``a : A